Praha- Dubna Praha- Dubna SPIN2013 SPIN2013 Helical magnets Siberian snakes I.Koop, A.Otboyev, P.Shatunov Yu.Shatunov Budker Institute for Nuclear Physics Novosibirsk
Praha- Dubna Praha- Dubna
SPIN2013SPIN2013
Helical magnets Siberian snakes
I.Koop, A.Otboyev, P.Shatunov
Yu.Shatunov
Budker Institute for Nuclear PhysicsNovosibirsk
Helical magnet
yoke
cm
cm
coil
yoke
coil
Transverse cross section of the field map
Bz
kGs
cm
cm
Bx
kGs
cm
cm
By
kGs
cm
cm
Helix field components on the axis (λ=2.5 m)
Bx
By
Particle and spin motion equations in the Cartesian frame
(Bρ is a rigidity)
Field in helical magnet
32
0 1 2
(3 )(2 )22 cos sin 2 2 cos3 ..
3( , )
;
I krI krb r b b
kr krr
B grad scalar potential
2 2
2 3
0 2
32 (1 )cos cos3
8 81;
k rb r b k rkr
2 2
2 2
0
2 2
2 2
0
2
2 2
0
{[1 (3 ) sin( cos( )]},8 4
{[1 ( 3 ) cos( ) sin( )]},8 4
{1 ( )}[ cos( ) sin( )].8
x
y
k kB b x y kz xy kz
k kB b x y kz xy kz
kBz b x y x kz y kz
Orbit in helical magnet (zero approximation)
0 0sin ; cos ; 0.x y zB b kz B b kz B
0 0
0 0
0 00[1 cos ] ;
| |
sin ( )| |
; ;| |
px kz x x p
k
p py kz y R y
k
q b eq
c k m
0 0
0 0
0
0
x y
x y
x
y
Spin in helical magnet (zero approximation)
1
2
3 00
cos sin ;
cos sin ;
;
x y
y x
z
e e kz e kz
e e kz e kz
qe e a
q
e3
e1
e2
S
-k
0
| |(1 )
p k
2 32 2
2 2
1( Re );
1
11 ;s
kn Ape RkA p
A p A a
For protons (a=1.793) p=1 by b0λ=19.6 Tm
Siberian snakes and spin rotators
1.Spin rotation2.No orbit disturbing and coupling outside
α1 α2α3 α4
R1 R2R3 R4
p1 p2p3 p4
R1=R4; R2 =R3
p1=-p4; p2 =-p3
snakes rotators
.sin 0
cos 0i i i
i i i
p R
p R
Siberian snakes and spin rotatorsfor RHIC (field)
Siberian snakes and spin rotatorsfor RHIC (orbit E=25 GeV)
Siberian snakes and spin rotatorsfor RHIC (spin)
Siberian Snake in RHIC4 superconducting helical dipoles:
Magnetic field 4T, length 2.4 m each with 360° twist, coil inner aperture 100 mm.
RHIC polarization
E=255 GeVL=5·1031cm-2s-1
S~50%
Snake from 2 helical magnets
BxBy
ξ = + ξ = -
z (cm)
Optimal particle trajectory
y
x
z (cm)
(cm
)
Spin trajectory S(0)=Sy→ -Sy
z (cm)
Sy
Sx
Sz
Spin trajectory
z (cm)
Sz
Sx
Sy
S(0)=Sz→ -Sz
Partial snakes
Partial snakes(field on axis)
Helix 3.4 m (λ=0.75 m)
correctorcorrector
Proton’s trajectory in the snake
x
E=25 ГэВ
Helix 3.4 m (λ=0.75 m)
correctorcorrector
Spin in partial snake (33%)
ACCELERATION OF POLARIZED PROTONS IN THE AGS WITH TWO
HELICAL PARTIAL SNAKES
H. Huang, L.A. Ahrens, M. Bai, K. Brown, E. D. Courant, C. Gardner, J.W. Glenn, R. C. Gupta,A.U. Luccio, W.W. MacKay, V. Ptitsyn, T. Roser, S. Tepikian, N. Tsoupas, E. Willen, A. Zelenski,K. Zeno, BNL, Upton, USA M. Okamura, J. Takano, Radiation Laboratory, RIKEN, Saitama, Japan, F. Lin, Indiana University, Bloomington
13%
6%AGS
S=70%
Partial snakes at U-70
Partial snakes at U-70(spin tune)
NICA polarization?
NICA polarization
(protons 10 GeV)
Helical magnet snakeB=4 T; L=10 m
Δx~Δz~1-2 mmSolenoid snakeB=4T; L=10 m
(coupling?)
l r
⊗⊙
10 sec
10e cool
E GeV
5 6 7 8 9 10 11 12 13 14 150
0.016
0.032
0.048
0.064
0.08
0.096
0.11
0.13
0.14
0.16
Kinetic Energy, GeV
IBS
Diffu
sio
n R
ate
s, s^(-
1)
transverse
longitudinal
IBS
diff
usio
n ra
te
(s-1)
NICA polarization + luminosity
Le
1, ,
0 1x z z
LT T T
“Rotating” quads
ang
le
I1/I2
Luminosity considerations
Coulomb scattering cross-section:
2
2 4 2max
12 μbarnpCoulomb
r
max( 5 mrad)
Limitations:
●
●
space-charge effect 20.1
2 2 ( 1)p p
p
b s p
N r R
n
instabilities in electron cooler:110.8 10bN
● beam-beam effect 0.034
p pp
p p p
N r
n
12bn bunches.
0 30 cms Assumptions:
●
●●
●
121 2 1 10N N
round beams
1 2 0
1 22 ( )b
N N fL
n
electron cooling will squeeze beams to the space charge limit ●
Luminosity considerations
Ek (GeV)
Np=1011
Conclusion
Thanks for attention!
Let’s do it
!